Is it possible to change the VPC of an already created EKS cluster? Or do I have to create a new one and there to select the new VPC?
you should be able to change the VPC configuration for the EKS cluster. However, as per the documentation which I found, it states that if VPC config is updated, the update type is replacement i.e., a new cluster will be created with the updated config.
Please see https://docs.aws.amazon.com/AWSCloudFormation/latest/UserGuide/aws-resource-eks-cluster.html#cfn-eks-cluster-resourcesvpcconfig for more information
Hope this helps.